Maxwell’s Equations
James Clerk Maxwell (1831-1874), a Scottish mathematician, studied properties
of electric and magnetic fields and came up with a set of 20 partial differential
equations in 20 unknowns which described mathematically how electric and magnetic
fields interact. Much later, an English electrical engineer by the name of Oliver
Heaviside (1850-1925), greatly simplified Maxwell’s equations to four equations in
two unknowns. A modern day version of the Maxwell equations in SI units^8 are
∇·E= ρ
0∇× E=−∂B
∂t
∇·B=0∇× B=μ 0 J+μ 0
0∂E
∂t(9 .134)In the Maxwell equations (9.134) one finds the following quantities
E=E(x, y, z, t) Electric field intensity (N/coul)
J=J(x, y, z ) Total current density (amp /m^2 )
B =B(x, y, z, t) Magnetic field intensity (N/amp ·m)
ρ=ρ(x, y, z) charge density (coul/m^3 )
μ 0 =4π× 10 −^7(N
amp^2
)the permeability of free space
0 =8. 85 × 10 −^12(coul^2
N·m^2
)the permittivity of free space
It is left as an exercise to show that the Maxwell equations are dimensionally homo-
geneous.
Note 1: Warning! The symbols B and H occur in the study of electromagnetism.
The symbol H is used to denote magnetic fields within a material medium. It
has no name, but some textbooks call it a magnetic induction– which is wrong.
To make matters worse many textbooks interchange the roles of B and H. My
only suggestion is be aware of these conflicts and study any textbook carefully
and see how things are defined.
(^8) There are two popular sets of units used to represent the Maxwell equations. These two popular units are the
International System of Units or Syst`eme international d ́unites, designatedSI (mks) in all languages and the
Gaussian (cgs) set of units. The main advantage of the Gaussian units is that they simplify many of the basic
equations of electricity and magnetism more so than the SI units.